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Models with limited dependent variables. Doctoral Program 2006-2007 Katia Campo. 3. Nested Logit Model. (K.Train, Ch.4, Franses and Paap Ch.5). 3. Nested Logit Model. Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern Example:.

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Models with limited dependent variables

Models with limited dependent variables

Doctoral Program 2006-2007

Katia Campo


3 nested logit model

3. Nested Logit Model

(K.Train, Ch.4, Franses and Paap Ch.5)


3 nested logit model1
3. Nested Logit Model

  • Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern

  • Example:

Choice of transportation mode

Car

Transit

Carpool

Bus

Train

Car alone


3 nested logit model2
3. Nested Logit Model

  • Assumption cumulative distribution of error terms

  • Probability


3 nested logit model3
3. Nested Logit Model

  • Probability can be decomposed into 2 logit models

(1)

(2)


3 nested logit model4
3. Nested Logit Model

  • Link between Pni | Bk * Pn Bk (upper and lower model level) through inclusive value Ink

  • To comply with RUM k must be in the [0,1] interval

  • IIA within, not across nests


3 nested logit model5
3. Nested Logit Model

Estimation

  • Joint estimation

  • Sequential estimation

    • Estimate lower model

    • Compute inclusive value

    • Estimate upper model with inclusive value as explanatory variable

      Disadvantages sequential estimation

    • Add fourth step, using the parameter estimates as starting values for joint estimation


3 nested logit model6
3. Nested Logit Model

  • Example: Location choices by French firms in Eastern and Western Europe

  • Dependent var.: probability of choosing location j Pj=P(πj> πk  k≠j )

  • Location choices are likely to have a nested structure (non-IIA)

    • First, select region (East or Western Europe)

    • Next, select country within region

Disdier and Mayer (2004)


3 nested logit model7
3. Nested Logit Model

  • Example 1: Location choices by French firms in Eastern and Western Europe

Location choice

E.Eur

W.Eur

………

………

CN

C1

CJ+1

CJ

Disdier and Mayer (2004)


3 nested logit model8
3. Nested Logit Model

  • Location choices: Data

    • 1843 location decisions in Europe from 1980 to 1999 (official statistics)

    • 19 host countries (13 W.Eur, 6 E.Eur)

Disdier and Mayer (2004)


3 nested logit model9
3. Nested Logit Model

  • Location choices: Data

    • NF French firms already located in the country

    • GDP GDP

    • GPP/CAP GDP per capita

    • DIST Distance France – host country

    • W Average wage per capita (manufacturing)

    • UNEMPL unemployment rate

    • EXCHR Exchange rate volatility

    • FREE Free country

    • PNFREE Partly free and not free country

    • PR1 Country with political rights rated 1

    • PR2 Country with political rights rated 2

    • PR345 Country with political rights rated 3,4,5

    • PR67 Country with political rights rated 6,7

    • LI Annual liberalization index

    • CLI Cumulative liberalization index

    • ASSOC =1 if an association agreement is signed

Disdier and Mayer (2004)


3. Nested Logit Model

Disdier and Mayer (2004)


3. Nested Logit Model

Disdier and Mayer (2004)


3. Nested Logit Model

  • Example 2: Shopping centre choice


3. Nested Logit Model

  • Example 2: Shopping centre choice


3. Nested Logit Model

  • Example 2: Shopping centre choice


3 nested logit model10
3. Nested Logit Model

  • (Purchase) incidence and choice models can be linked in the same way

Include incl.value 

Purchase decision

No purchase

Purchase

...

(See above: Bucklin & Gupta)

Altern. J

Altern.1


3 nested logit model11
3. Nested Logit Model

  • Other examples

    • Private label versus nationals brands

    • Same versus other brand

    • Fixed rate (low risk) versus variable rate (high risk) investments

    • Choice of transportation mode

    • ....


4 probit model

4. Probit Model

(K.Train, Ch.5 ; Franses and Paap, Ch.4-5)


4 probit model1
4. Probit Model

  • Based on the general RUM-model

  • Ass.: error terms are distributed normal with a mean vector of zero and covariance matrix 

  • density function:


4 probit model2
4. Probit Model

Logit or Probit?

  • Trade-off between tractability and flexibility

    • Closed-form expression of the integral for Logit, not for Probit models

    • Probit allows for random taste variation, can capture any substitution pattern, allows for correlated error terms and unequal error variances

 Dependent on the specifics of the choice situation


4 probit model3
4. Probit Model

Estimation

  • Approximation of the multidimensional integral

    • Non-simulation procedures (see Kamakura 1989)

      Can usually only be applied to restricted cases and/or provide inaccurate estimations

    • Simulation procedures (see Geweke et al.1994, Train)


4 probit model4
4. Probit Model

Simulation-based estimation(binary probit, CFS)

  • Step 1

    • For each observation n=1, ..., N draw r ~ N(0,1), (r = 1, ......., R: repetitions)

    • Initialize y_count= 0, =mt (starting values)

    • Compute y*rn = xn mt + L r ; L= choleski factor (LL’= )

    • Evaluate: y*rn >0  y_count= y_count+1

    • Repeat R times

Weeks (1997)


4 probit model5
4. Probit Model

Simulation-based estimation(binary probit, CFS)

  • Step 2: calculate probabilities

    Pn| mt= y_count/R

  • Step 3: Form the simulated LL function

    SLL=  n yn ln(Pn|mt)+(1-yn) ln(1-Pn|mt)

  • Step 4: Check convergence criteria (SLL(mt)- SLL(mt-1))

  • Step 5: Update mt: mt+1 = mt + v

  • Step 6: Iterate (until convergence)

Weeks (1997)


4 probit model6
4. Probit Model

Simulation-based estimation: MNProbit

  • Based on same principles

  • More efficient simulation procedures (see Train)

  • Identification: normalization of level and scale

    • Re-express model in utility differences

    • Normalization of varcov matrix (see Train)


4 probit model7
4. Probit Model

  • Random taste variation

  • Model with random coefficients

    • E.g.: n~N(b,W)

    • Unj = b’xnj+ *’n xnj + nj

    • = b’xnj+ nj

    • nj : correlated error terms (dep.on xnj, see Train)


4 probit model8
4. Probit Model

  • Substitution patterns

  • Full covariance matrix (no parameter restrictions) unrestricted substitution patterns

  • Structured covariance matrix (restrictions on some covariance parameters)

  •  the structure imposed on  determines the substitution pattern and may allow to reduce the number of parameters to be estimated


4 probit model9
4. Probit Model

  • Example (Kamakura and Srivastava 1984): random utility components ni, nj are more (less) highly correlated when i and j are more (less) similar on important attributes

(dij = weighted eucledian distance between i & j)


4 probit model10
4. Probit Model

  • Examples

    • Choice models at brand-size level: correlation between ≠ sizes of same brand (Chintagunta 1992)

MNL model

gives biased

estimates

of price

elasticity


4 probit model11
4. Probit Model

  • Examples

    • Firm innovation (Harris et al. 2003)

      • Binary probit model for innovative status (innovation occurred or not)

      • Based on panel data  correlation of innovative status over time: unobserved heterogeneity related to management ability and/or strategy



4 probit model12
4. Probit Model

  • Examples

    • Dynamics of individual health (Contoyannis, Jones and Nigel 2004)

      • Binary probit model for health status (healthy or not)

      • Survey data for several years  correlation over time (state dependence) + individual-specific (time-invariant) random coefficient


4 probit model13
4. Probit Model

  • Examples

    • Choice of transportation mode (Linardakis and Dellaportas 2003)  Non-IIA substitution patterns




5 ordered logit model1
5. Ordered Logit Model

  • Choice between J>2 ordered ‘alternatives’

  • Ordinal dependent variable y = 1, 2, ... J, with

    rank(1) < rank(2) < ... < rank(J)

  • Example:

    • Purchase of 1, 2, ... J units

    • Evaluation on a J-point scale ranging from, e.g., ‘dislike very much’ to ‘like very much’


5 ordered logit model2
5. Ordered Logit Model

  • Suppose yi* is a continous latent variable which

    • is a linear function of the explanatory variables

      yn* = Xn + n

    • and can be ‘mapped’ on an ordered multinomial variable as follows:

      yn= 1 if 0 < yn*  1

      yn= j if j-1 < yn*  j

      yn= J if J-1 < yn*  J

      0 < 1 < …. < j < … < J


5 ordered logit model3
5. Ordered Logit Model

Ordered logit (see above)

  • 0 , J and 0: set equal to zero


5 ordered logit model4
5. Ordered Logit Model

Interpretation of parameters (marginal effects)


5 ordered logit model5
5. Ordered Logit Model

Estimation: ML


5 ordered logit model6
5. Ordered Logit Model

  • Disadvantages (Borooah 2002)

  • Assumption of equal slope k

  • Biased estimates if assumption of strictly ordered

    outcomes does not hold

  • treat outcomes as nonordered unless there are

    good reasons for imposing a ranking


5 ordered logit model7
5. Ordered Logit Model

Example: Effectiveness of better public transit as a way to reduce automobile congestion and air polution in urban areas

  • Research objective: develop and estimate models to measure how public transit affects automobile ownership and miles driven.

  • Data: Nationwide Personal Transportation Survey (42.033 hh): socio-demo’s, automobile ownership and use, public transportation avail.

Kim and Kim (2004)


5 ordered logit model8
5. Ordered Logit Model

  • Dependent variable ownership model = number of cars (k = 0, 1, 2,  3)  ordinal variable

  • C*i = latent variable: automobile ownership propensity of hh i

  • Relation to observed automobile ownership:

    Ci=k if k-1 < ’xi +  < k

  • P(Ci=k)=F(k- ’xi) - F(k-1 - ’xi)

Kim and Kim (2004)





5. Ordered Logit Model

  • Examples

  • Occupational outcome as a function of socio-demographic characteristics (Borooah)

    • Unskilled/semiskilled

    • Skilled manual/non-manual

    • Professional/managerial/technical

  • School performance (Sawkins 2002)

    • Grade 1 to 5

    • Function of school, teacher and student characteristics

  • Level of insurance coverage


D heterogeneity
D.Heterogeneity

  • Observed heterogeneity

  • Unobserved heterogeneity

    • Over decision makers

      • Random coefficients Models

      • E.g. Mixed Logit Model (see Train)

    • Over segments

      • Latent class estimation


D heterogeneity latent class est
D.Heterogeneity: Latent class est.

  • Ass.: Consumers can be placed into a small number of – homogeneous - segments which differ in choice behavior ( response parameters)

  • Relative size of the segment s (s=1, 2, ..., M) is given by

    fs = exp(s) / s’exp(s’)

Kamakura and Russell (1989)


D heterogeneity latent class est1
D.Heterogeneity: Latent class est.

  • Probability of choosing brand j, conditional on consumer i being a member of segment s

    Ps(yn = j|Xn) = exp(Xjns)

    lexp(Xlns)


D heterogeneity latent class est2
D.Heterogeneity: Latent class est.

  • Unconditional probability that consumer i will choose brand j

    P(yn = j|Xn) = sfs Ps(yn = j|Xn)

    = s exp(s) exp(Xjns)

    s’exp(s’) lexp(Xlns)


D heterogeneity latent class est3
D.Heterogeneity: Latent class est.

  • Estimation: Maximum Likelihood

  • Likelihood of a hh’s choice history Hn

    L(Hn) = s [ exp(s)L(Hn|s) / s’ exp(s’) ]

    with

    L(Hn|s) = t Ps(ynt = c(t) | Xnt)

    c(t) = index of the chosen option at time t.

  • Maximize likelihood over all hh’s: n L(Hn)


D heterogeneity latent class est4
D. Heterogeneity: Latent class est.

Segment analysis

  • Based on parameter estimates (e.g. difference in price sensitivity)

  • Based on segment profiles

    • Post-hoc: based on assignment of hh to segments; Probability that hh n belongs to segment s =

      P(ns | Hn) = L(Hn|s)fs / s’ [L(Hn|s’)fs’]

      Analyze characteristics of different segments

    • A priori: make fs a function of variables that may explain segment membership (e.g. income for segments which differ in price sensitivity)


D heterogeneity latent class est5
D.Heterogeneity: Latent class est.

  • Example: heterogeneity in price sensitivity (Bucklin and Gupta 1992)


D heterogeneity latent class est6
D.Heterogeneity: Latent class est.

  • Example: heterogeneity in price sensitivity



D heterogeneity latent class est8
D.Heterogeneity: Latent class est.

Choice segments: segment 1 = more sensitive to price and promo

Incidence segments: segment 2 and 4 = more sensitive to changes

in category attractiveness (change in price/promo)

 Confirms that  combinations of choice/inc.price sensitivity occur



D heterogeneity latent class est10
D. Heterogeneity: Latent class est.

  • Segment analysis: price elasticity


D heterogeneity latent class est11
D. Heterogeneity: Latent class est.

  • Segment analysis: socio-demographic profile


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